Unravelling the large deviation statistics of Markovian open quantum systems

We analyse dynamical large deviations of quantum trajectories in Markovian open quantum systems in their full generality. We derive a {\em quantum level-2.5 large deviation principle} for these systems, which describes the joint fluctuations of time-averaged quantum jump rates and of the time-averaged quantum state for long times. Like its level-2.5 counterpart for classical continuous-time Markov chains (which it contains as a special case) this description is both {\em explicit and complete}, as the statistics of arbitrary time-extensive dynamical observables can be obtained by contraction from the explicit level-2.5 rate functional we derive. Our approach uses an unravelled representation of the quantum dynamics which allows these statistics to be obtained by analysing a classical stochastic process in the space of pure states. For quantum reset processes we show that the unravelled dynamics is semi-Markov, and derive bounds on the asymptotic variance of the number of quantum jumps which generalise classical thermodynamic uncertainty relations. We finish by discussing how our level-2.5 approach can be used to study large deviations of non-linear functions of the state such as measures of entanglement.